On PI-control in Capacity-Limited Networks

Anders Rantzer; Felix Agner

REVIEW 5 major objections 5 minor 29 references

On PI-control in Capacity-Limited Networks

T0 review · 5 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A fully decentralized anti-windup PI controller drives capacity-limited networks to a unique globally stable equilibrium that minimizes a weighted sum of tracking errors, while a rank-1 coordinated variant minimizes the worst-case error.

desk verdict Strong nonlinear extension of earlier anti-windup results, but the core proofs currently contain sign errors that need repair before the stability and optimality claims are established. read the letter →

arxiv 2411.14077 v1 pith:7RBWMPEV submitted 2024-11-21 eess.SY cs.SY

Felix Agner , Anders Rantzer This is my paper · ORCID

classification eess.SYcs.SY

keywords anti-windupcontrolPImulti-agentsystemscapacity-limitednetworksglobalasymptoticstabilityequilibriumoptimalitydistrictheatingnonlinearinterconnection

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes that two simple anti-windup PI controllers can optimally share a limited resource among many agents, even though the coupling between agents is unknown and nonlinear. In the fully decentralized version, each agent uses only local measurements and fixed gains; the closed loop has a unique, globally asymptotically stable equilibrium, and this equilibrium beats every other open-loop equilibrium with a different saturated input in a weighted sum of absolute tracking errors. Adding a single rank-1 coordination signal to the anti-windup makes any closed-loop equilibrium minimize the largest tracking error across agents. These theorems rest only on structural assumptions about the coupling: competition between agents and a positive weighted monotonicity. A district-heating case study shows how the model covers real capacity constraints, and simulations indicate both controllers track their corresponding optimal equilibria.

What carries the argument

The load-bearing object is the nonlinear coupling map $b:\mathcal{S}\to\mathcal{B}$ in the agent dynamics, where $\mathcal{S}$ is the saturation range. Assumption 1 gives it a competition property (if $v\ge \hat v$ and $v_i = \hat v_i$, then $b_i(v)-b_i(\hat v)<0$) and a positive weighted monotonicity ($\eta^\top(b(v)-b(\hat v))>0$ for $v\ge \hat v$, $v\ne \hat v$). From these, Lemma 1 and Lemma 2 extract the sign-dominance and inverse-monotonicity features used in every proof. Stability of the decentralized loop is carried by the weighted $\ell^1$ Lyapunov function $V=\sum_i \eta_i \frac{d_i}{p_i}|\tilde\zeta_i|+\eta_i|\tilde u_i|$; optimality arguments compare any other open-loop equilibrium to the saturated control $u_0$ and use the competition structure to force a strict inequality. The coordinating controller's anti-windup term $k_C \mathbf{1}\mathbf{1}^\top \mathrm{dz}(u)$ is what forces $x$ to be parallel to $\mathbf{1}$, which is the source of the min-max property.

What would settle it

Run the decentralized controller on a two-agent plant whose coupling satisfies Assumption 1, then enumerate all open-loop equilibria and check whether any one has a strictly smaller $\sum_i \eta_i a_i |x_i|$ than the closed-loop equilibrium; a single such case would disprove Theorem 2. Alternatively, in a district-heating test rig hold one valve fixed while opening all others and measure that consumer's flow: if the flow does not strictly drop, Assumption 1(i) fails and the theory's application to that network is invalid.

Watch

Extended reading notes

Core claim

The paper's central claim is that global stability and equilibrium optimality survive a full nonlinear generalization of capacity-limited resource sharing. For a plant of $n$ agents with dynamics $\dot x_i = -a_i x_i + b_i(\mathrm{sat}(u)) + w_i$ and a bounded nonlinear coupling $b$ satisfying Assumption 1, the decentralized anti-windup PI controller (2) has a unique globally asymptotically stable equilibrium (Theorem 1), and this equilibrium strictly minimizes $\sum_i \eta_i a_i |x_i|$ among all open-loop equilibria whose saturated input differs (Theorem 2). The rank-1 coordinating controller (4) has the property that every closed-loop equilibrium satisfies $x = -k_C \mathbf{1}\mathbf{1}^\top \mathrm{dz}(u)$, so all agents share the same error, and any such equilibrium is the unique minimizer of the $\infty$-norm of the tracking error among open-loop equilibria with different saturated input (Theorem 3). Global asymptotic stability for the coordinating controller is proved only when the disturbance can be fully rejected (Theorem 4); in the saturated regime existence and stability are not guaranteed.

Load-bearing premise

The single most load-bearing premise is Assumption 1 on the otherwise unknown coupling map: increasing another agent's input must lower your resource share, and some positive weighted sum of all shares must rise when everyone increases input; if a physical network lacks this competitive monotone structure, the stability and optimality conclusions do not follow.

Editorial extensions

If this is right

A decentralized anti-windup PI controller can be installed per agent with local gains satisfying $k_i^P a_i > k_i^I$ and $k_i^P k_i^A < 1$; it will globally stabilize the network and reach an equilibrium that is optimal in the weighted absolute-error sense, with no communication and no model of the coupling.
The decentralized equilibrium's optimality is strict: any other open-loop equilibrium with different saturated inputs has a strictly larger value of $\sum_i \eta_i a_i |x_i|$, so the controller is doing implicit real-time optimization.
The rank-1 coordinating controller forces all agents to share the same tracking error at equilibrium, and any such equilibrium is strictly better than every other open-loop equilibrium in the infinity-norm sense, giving an explicit fairness and worst-case guarantee.
Global asymptotic stability for the coordinating controller is proved only when the disturbance is small enough to reject; for larger disturbances, the paper guarantees equilibrium optimality but leaves existence and stability of the saturated coordinating equilibrium open.
In district heating, the decentralized and coordinating laws reproduce the offline optimal allocations for average and worst-case temperature deviations respectively, suggesting that simple valve-level PI control can replace centralized optimization in practice.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One consequence the authors leave implicit is that when Assumption 1 holds for a whole family of weight vectors $\eta$, the fully decentralized controller simultaneously optimizes all the corresponding weighted absolute-error costs, giving a multi-objective optimality statement beyond any single chosen $\eta$.
The proof machinery suggests a model-free design recipe: verify only the competition and weighted monotonicity of the steady-state input-output map, then tune local PI and anti-windup gains independently; this could be tested in other saturated resource networks such as communication or building cooling systems.
Because the coordination signal is rank-1, it can be implemented by a single broadcast channel; a natural extension would replace the fixed vector $\mathbf{1}$ by any positive weight vector to obtain weighted min-max equilibria, something the paper does not explore.
In the saturated regime, the coordinated controller's existence and stability remain open; a concrete next step is to search for a Luapunov function or a counterexample in the two-agent saturated case, which would settle whether the min-max equilibrium is actually reachable.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper is a genuine nonlinear generalization of the authors' earlier M-matrix anti-windup PI results, and the main theorems are plausible—but the proofs as written contain sign/inequality errors in key steps. The paper needs correction before the central stability and optimality claims can be taken as established.

What is actually new: the interconnection b is allowed to be fully nonlinear, with only a monotone-competition assumption (Assumption 1) instead of a linear M-matrix structure. On that basis, Theorems 1 and 2 show that a fully decentralized anti-windup PI controller yields global asymptotic stability and a weighted-l1 optimal equilibrium, and Theorem 3 gives l∞ optimality for a rank-1 coordinated variant. The district heating example is a nice real-world motivation, and the fact that no model of b is needed is a real strength.

The soft spots are real. The stress-test note is correct. In the proof of Theorem 1, term (13d) is bounded via (14) with '≥' and a plus sign in (15), but the correct bound has the opposite sign; Lemma 1 then gives strict negativity, but the printed chain does not. In the proof of Theorem 2, Eq. (28) asserts sign(dz_i(u0)) = sign(v†_i - v0_i), which is backwards; it should be sign(v0_i - v†_i). The later expression in (32) uses the correct sign, so the proof is likely repairable, but as printed the derivation is invalid. These are not minor typos in symbols only—they break the logical connection in the two main theorems. In addition, Proposition 1 (hydraulic monotonicity) is unproved and load-bearing for the simulation, Lemma 3 is only sketched, and the Lyapunov function uses a nonsmooth convention that is acknowledged but not fully resolved. Theorem 3's caveat about equilibrium existence is stated honestly.

All that said, the paper deserves a serious referee. The framework is interesting, the assumptions are clean, and the results, if corrected, would be a solid contribution. I would recommend a revise-and-resubmit with the request to fix the sign errors, provide a proof or reference for Proposition 1, and tighten the technical lemmas. I would not cite it in its current form.

Referee Report

5 major / 5 minor

Summary. The paper considers a class of multi-agent systems in which each agent has scalar first-order dynamics ẋ_i = −a_i x_i + b_i(sat(u)) + w_i, with a nonlinear, bounded, monotone-competition interconnection b satisfying Assumption 1. For a fully decentralized PI controller with anti-windup (2), Theorem 1 claims global asymptotic stability of a unique equilibrium, and Theorem 2 claims that this equilibrium minimizes the weighted l1 cost Σ η_i a_i |x_i| among all open-loop equilibria with different saturated input. For a rank-1 coordinating anti-windup variant (4), Theorem 3 claims that any equilibrium minimizes the l∞ tracking error, and Theorem 4 establishes global asymptotic stability in the disturbance-rejection regime b(l)+w > 0 and b(l̄)+w < 0. The paper also presents a district heating motivating example, a numerical simulation, and a proposition (Proposition 1) asserting that the hydraulic flow map satisfies Assumption 1.

Significance. If the results hold, the paper meaningfully extends the linear M-matrix results of [18,19] to a broad nonlinear setting, requiring only qualitative monotonicity properties rather than an explicit model of the interconnection. The proofs are based on a coherent Lyapunov framework and a key inequality (Lemma 1), and the optimality claims are derived from the assumptions rather than fitted to pre-selected cost functions. The authors are honest about the main limitations: Theorem 3 is conditional on equilibrium existence, the global stability proof for the coordinating controller covers only the unsaturated regime, and Proposition 1 is stated without proof. These are real gaps, but the core ideas are attractive and likely correct after the proof issues below are fixed.

major comments (5)

[§5.2, Eq. (14)–(16)] The inequality direction used to bound term (13d) is reversed. The text states that (13d) is bounded below by (15)–(16) via '≥' and then concludes that the expression is strictly negative; however, Lemma 1 gives sign( ˜sat (˜u))^T H ˜b( ˜sat (˜u)) > Σ_{j∈J0} η_j |˜b_j|, so the right-hand side of (15)–(16) is strictly positive when ˜sat ≠ 0. What is needed for negative definiteness of ˙V is an upper bound, i.e., (13d) ≤ −(15)+(16), with the opposite inequality. This is a load-bearing error in the proof of Theorem 1 and must be corrected before the global asymptotic stability claim is established.
[§5.3, Eq. (28)] The identity sign(dz_i(u0)) = sign(v†_i − v0_i) is incorrect. Saturation logic gives sign(dz_i(u0)) = sign(v0_i − v†_i): if dz_i(u0) > 0 then v0_i is the upper saturation bound and v†_i < v0_i, while if dz_i(u0) < 0 then v0_i is the lower bound and v†_i > v0_i. The final expression in (32) uses sign(v0 − v†), so the sign in (28) must be reversed and the intermediate inequalities (27)–(29) revised accordingly.
[§5.2, Lemma 3] The existence of an equilibrium is essential for Theorem 1, but the proof of Lemma 3 is only a sketch. The map T is claimed to be forward-invariant on a sufficiently large box C, and then Brouwer's fixed-point theorem is invoked, but no argument is given for forward invariance or for the choice of α. A complete proof, or a precise reference, is required for this lemma.
[§4, Proposition 1] Proposition 1 asserts that the hydraulic flow map q(v) satisfies Assumption 1, and the text states 'we can show the following' and 'Hence q satisfies Assumption 1', but the proof is omitted. This proposition is load-bearing for the district heating application and the simulation: without it, the numerical example is not connected to the theoretical results. It should either be proved in an appendix with the hydraulic model stated in detail, or explicitly formulated as an assumption on the simulation model.
[§5.2, Eqs. (11)–(12)] The Lyapunov function (11) is not differentiable at points where a component of ˜ζ or ˜u is zero, and the proof uses the convention d/dt |x| = sign(x) ẋ as if it were an exact derivative. The brief remark about exchanging |·| by an arbitrarily close smooth approximation does not by itself justify the global asymptotic stability conclusion. A rigorous treatment via, e.g., Clarke generalized derivatives or an explicit smoothing argument with uniform negative-definiteness estimates is needed.

minor comments (5)

[§Appendix, Eq. (25)] The equivalence '2/kC − 1^T P 1 ≥ 0 ⇔ kC/2 1^T P 1 = kC/2 1^T kP ≥ 0' appears to contain a typo: the last expression should be '≤ 1', not '≥ 0'.
[§3.2, Assumption 3] The condition 'kC 2 1⊤kP ≤ 1' is ambiguous; it should be written as (kC/2) 1^T kP ≤ 1.
[§5.2, Lemma 4 proof] The assertion that dz_i(u) > 0 implies b_i(sat(u)) + w_i > 0 via Assumption 1(i) is not immediate because sat_i(u) is then the upper bound, so Assumption 1(i) does not directly apply unless one first derives monotonicity of b_i in its own input from the combination of Assumption 1(i)–(ii). A short justification would make this step clear.
[§1.1, Notation] The use of the same symbol 'l' for the lower and upper saturation bounds (in the definition of S and later in Assumption 3 and Lemma 4) is confusing; consider using l_low and l_high or l_i and l̄_i for clarity.
[§4.1, Figure 3] The simulation comparison is qualitative. Reporting quantitative error metrics (e.g., final values of ||x||_1 and ||x||_infinity in each scenario) would make the illustrative comparison more informative.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: all central claims are derived from explicit assumptions without fitted parameters or load-bearing self-citation.

full rationale

The paper's main results (Theorems 1, 2, and 3) are obtained by explicit Lyapunov and comparison arguments from Assumptions 1, 2, and 3. No parameter is fitted to the quantity later called a prediction: the controller gains are only constrained by inequalities (Assumption 2 and Assumption 3), and they are not tuned to the cost functions eta_i a_i |x_i| or ||x||_infinity. The optimality costs emerge from the equilibrium equations and from Lemma 1, which is proved from Assumption 1 rather than assumed as the desired conclusion. The district-heating simulation relies on a separate hydraulic model; Proposition 1 is stated as an omitted technical proof, not as a consequence of the controller, and the benchmarks independently minimize the same costs via convex optimization, so the controllers are not calibrated to the benchmarks. Citations to the authors' prior work [18,19] are contextual (the linear special case) and are not used to establish the nonlinear theorems; existence of the coordinating equilibrium is explicitly left open and referenced only for the linear case. The sign and inequality issues identified in the proofs of Theorems 1 and 2, if confirmed, are correctness defects rather than circular reductions, because they do not make the theorem equivalent to its assumptions by construction. Therefore the derivation chain is self-contained and no circularity is present.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no fitted free parameters: the controller gains kP, kI, kA, kC are fixed design choices, and the weight vector eta is part of the assumption on b, not estimated from data. The main theoretical results rest on the monotonicity assumptions A1-A3, which are natural but unverified for real networks. The district heating application adds an unproved assertion (Proposition 1) that the hydraulic flow map satisfies A1.

assumptions (4)

domain assumption Assumption 1: The interconnection b is continuous, and there exists eta > 0 such that (i) if v>=v and v_i = v_i, then b_i(v)-b_i(v)<0; (ii) eta^T(b(v)-b(v)) > 0 for any v>=v, v!=v.
This is the central structural assumption on the unknown coupling. It encodes competition (opening one valve reduces others' flow) and total throughput increase when all valves open. Without it, Theorems 1-4 do not hold.
domain assumption Assumption 2: For all i, kP_i a_i > kI_i and kP_i kA_i < 1.
Controller gain condition required for the decentralized Lyapunov proof to dominate the anti-windup term.
domain assumption Assumption 3: a_i kP_i = (1+alpha) kI_i with alpha>0, and kC/2 sum_i kP_i <= 1.
Gain condition for the coordinating controller; ensures Schur complement in Lemma 4 is positive semidefinite.
domain assumption Assumption 4: The temperature differences delta_i, cp,w, and rho_w are constant; the hydraulic network is tree-structured with a constant-capacity pump (Proposition 1).
Simplifications used to map district heating dynamics to the abstract model. Proposition 1, which asserts that the flow map q(v) satisfies Assumption 1, is stated without proof.

how reviews work

0 comments

Cite this review

Pith. "Pith review of On PI-control in Capacity-Limited Networks." pith.science (2026). https://pith.science/paper/7RBWMPEV

@misc{pith2026241114077,
  author       = {Pith},
  title        = {Pith review of: On PI-control in Capacity-Limited Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7RBWMPEV}},
  note         = {Machine review of arXiv:2411.14077}
}

read the original abstract

This paper concerns control of a class of systems where multiple dynamically stable agents share a nonlinear and bounded control-interconnection. The agents are subject to a disturbance which is too large to reject with the available control action, making it impossible to stabilize all agents in their desired states. In this nonlinear setting, we consider two different anti-windup equipped proportional-integral control strategies and analyze their properties. We show that a fully decentralized strategy will globally, asymptotically stabilize a unique equilibrium. This equilibrium also minimizes a weighted sum of the tracking errors. We also consider a light addition to the fully decentralized strategy, where rank-1 coordination between the agents is introduced via the anti-windup action. We show that any equilibrium to this closed-loop system minimizes the maximum tracking error for any agent. A remarkable property of these results is that they rely on extremely few assumptions on the interconnection between the agents. Finally we illustrate how the considered model can be applied in a district heating setting, and demonstrate the two considered controllers in a simulation.

Figures

Figures reproduced from arXiv: 2411.14077 by the authors.

**Figure 2.** Outdoor temperature To(t) used in simulation. optimally in each instance of the simulation as the solution to the problem minimize x, q J(x) (7a) subject to (6) with ˙xi = 0 for i = 1, . . . , n, (7b) q ∈ Q (7c) where we use J(x) = ∥x∥1 and J(x) = ∥x∥∞ respectively. This problem corresponds to calculating a flow rate q which is feasible within the hydraulic constraints of the network (i.e., (7b)) which generates an… view at source ↗

**Figure 3.** Resulting indoor temperatures. The three clusters of buildings, i.e., [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

29 extracted references · 27 canonical work pages

[1]

Rate control for communica- tion networks: shadow prices, proportional fairness and stability,

F. P. Kelly, A. K. Maulloo, and D. K. H. Tan, “Rate control for communica- tion networks: shadow prices, proportional fairness and stability,” Journal of the Operational Research Society , vol. 49, pp. 237–252, Mar. 1998

work page 1998
[2]

Optimization flow control. I. Basic algorithm and convergence,

S. Low and D. Lapsely, “Optimization flow control. I. Basic algorithm and convergence,” IEEE/ACM Transactions on Networking , vol. 7, pp. 861– 874, Dec. 1999

work page 1999
[3]

Internet congestion control,

S. Low, F. Paganini, and J. Doyle, “Internet congestion control,” IEEE Control Systems Magazine , vol. 22, pp. 28–43, Feb. 2002

work page 2002
[4]

Fairness and Stability of End-to-End Congestion Control*,

F. Kelly, “Fairness and Stability of End-to-End Congestion Control*,” Eu- ropean Journal of Control , vol. 9, pp. 159–176, Jan. 2003

work page 2003
[5]

Optimal power flow pursuit,

E. Dall’Anese and A. Simonetto, “Optimal power flow pursuit,” IEEE Transactions on Smart Grid , vol. 9, no. 2, pp. 942–952, 2018

work page 2018
[6]

A Survey of Distributed Optimization and Control Algorithms for Electric Power Systems,

D. K. Molzahn, F. D¨ orfler, H. Sandberg, S. H. Low, S. Chakrabarti, R. Baldick, and J. Lavaei, “A Survey of Distributed Optimization and Control Algorithms for Electric Power Systems,” IEEE Transactions on Smart Grid , vol. 8, pp. 2941–2962, Nov. 2017

work page 2017
[7]

Deployment of an Online Feedback Optimization Controller for Reactive Power Flow Optimization in a Distribution Grid,

L. Ortmann, C. Rubin, A. Scozzafava, J. Lehmann, S. Bolognani, and F. D¨ orfler, “Deployment of an Online Feedback Optimization Controller for Reactive Power Flow Optimization in a Distribution Grid,” in 2023 IEEE PES Innovative Smart Grid Technologies Europe (ISGT EUROPE) , (Grenoble, France), pp. 1–6, IEEE, Oct. 2023

work page 2023
[8]

A distributed algorithm for auto-balancing of hydronic cooling systems,

C. S. Kallesøe, B. K. Nielsen, A. Overgaard, and E. B. Sørensen, “A distributed algorithm for auto-balancing of hydronic cooling systems,” in 2019 IEEE Conference on Control Technology and Applications (CCTA) , pp. 655–660, 2019

work page 2019

Show all 29 references

[9]

Heat balancing in cooling systems using distributed pumping,

C. S. Kallesøe, B. K. Nielsen, and A. Tsouvalas, “Heat balancing in cooling systems using distributed pumping,” IF AC-PapersOnLine, vol. 53, no. 2, pp. 3292–3297, 2020. 21st IF AC World Congress

2020
[10]

Combating district heating bottlenecks using load control,

F. Agner, P. Kergus, R. Pates, and A. Rantzer, “Combating district heating bottlenecks using load control,” Smart Energy, vol. 6, 2022

2022
[11]

Control-Based Resource Management for Storage of Video Streams,

A. Martins, M. Lindberg, M. Maggio, and K.-E. ˚Arz´ en, “Control-Based Resource Management for Storage of Video Streams,”IF AC-PapersOnLine, vol. 53, no. 2, pp. 5542–5549, 2020

2020
[12]

Dynamic Management of Multiple Resources in Camera Surveillance Systems,

A. Martins and K.-E. ˚Arz´ en, “Dynamic Management of Multiple Resources in Camera Surveillance Systems,” in 2021 American Control Conference (ACC), (New Orleans, LA, USA), pp. 2061–2068, IEEE, May 2021. 20

2021
[13]

Real-Time optimization as a feed- back control problem – A review,

D. Krishnamoorthy and S. Skogestad, “Real-Time optimization as a feed- back control problem – A review,” Computers & Chemical Engineering , vol. 161, p. 107723, May 2022

2022
[14]

Optimiza- tion algorithms as robust feedback controllers,

A. Hauswirth, Z. He, S. Bolognani, G. Hug, and F. D¨ orfler, “Optimiza- tion algorithms as robust feedback controllers,” Annual Reviews in Control, vol. 57, p. 100941, 2024

2024
[15]

Model-free nonlinear feedback optimization,

Z. He, S. Bolognani, J. He, F. D¨ orfler, and X. Guan, “Model-free nonlinear feedback optimization,” IEEE Transactions on Automatic Control, vol. 69, no. 7, pp. 4554–4569, 2024

2024
[16]

The linear saturated decentralized strategy for constrained flow control is asymptotically opti- mal,

D. Bauso, F. Blanchini, L. Giarr´ e, and R. Pesenti, “The linear saturated decentralized strategy for constrained flow control is asymptotically opti- mal,” Automatica, vol. 49, no. 7, pp. 2206–2212, 2013

2013
[17]

Fair and sparse solutions in network-decentralized flow control,

F. Blanchini, C. A. Devia, G. Giordano, R. Pesenti, and F. Rosset, “Fair and sparse solutions in network-decentralized flow control,” IEEE Control Systems Letters, vol. 6, pp. 2984–2989, 2022

2022
[18]

Decentralized pi-control and anti-windup in resource sharing net- works,

F. Agner, J. Hansson, P. Kergus, A. Rantzer, S. Tarbouriech, and L. Zac- carian, “Decentralized pi-control and anti-windup in resource sharing net- works,” European Journal of Control , p. 101049, 2024

2024
[19]

Anti- windup coordination strategy around a fair equilibrium in resource sharing networks,

F. Agner, P. Kergus, A. Rantzer, S. Tarbouriech, and L. Zaccarian, “Anti- windup coordination strategy around a fair equilibrium in resource sharing networks,” IEEE Control Systems Letters , vol. 7, pp. 2521–2526, 2023

2023
[20]

A tutorial on modern anti-windup design,

S. Galeani, S. Tarbouriech, M. Turner, and L. Zaccarian, “A tutorial on modern anti-windup design,” European Journal of Control , vol. 15, no. 3, pp. 418–440, 2009

2009
[21]

On the robust implementation of projected dynamical systems with anti-windup controllers,

A. Hauswirth, F. D¨ orfler, and A. Teel, “On the robust implementation of projected dynamical systems with anti-windup controllers,” in 2020 Amer- ican Control Conference (ACC), pp. 1286–1291, 2020

2020
[22]

R. A. Horn and C. R. Johnson, Topics in Matrix Analysis . Cambridge University Press, 1991

1991
[23]

Consensus problems in networks of agents with switching topology and time-delays,

R. Olfati-Saber and R. Murray, “Consensus problems in networks of agents with switching topology and time-delays,” IEEE Transactions on Auto- matic Control, vol. 49, no. 9, pp. 1520–1533, 2004

2004
[24]

Output Regu- lation of Large-Scale Hydraulic Networks,

C. De Persis, T. N. Jensen, R. Ortega, and R. Wisniewski, “Output Regu- lation of Large-Scale Hydraulic Networks,” IEEE Transactions on Control Systems Technology, vol. 22, pp. 238–245, Jan. 2014

2014
[25]

On the Existence and Uniqueness of Steady State Solutions of a Class of Dynamic Hydraulic Networks via Actuator Placement,

M. Jeeninga, J. E. Machado, M. Cucuzzella, G. Como, and J. Scherpen, “On the Existence and Uniqueness of Steady State Solutions of a Class of Dynamic Hydraulic Networks via Actuator Placement,” in 2023 62nd 21 IEEE Conference on Decision and Control (CDC) , pp. 3652–3657, Dec

2023
[26]

Hydraulic parameter estimation for district heating based on laboratory experiments,

F. Agner, C. M. Jensen, A. Rantzer, C. S. Kallesøe, and R. Wisniewski, “Hydraulic parameter estimation for district heating based on laboratory experiments,” Energy, vol. 312, p. 133462, 2024

2024
[27]

Differentialequations.jl–a performant and feature-rich ecosystem for solving differential equations in julia,

C. Rackauckas and Q. Nie, “Differentialequations.jl–a performant and feature-rich ecosystem for solving differential equations in julia,” Journal of Open Research Software, vol. 5, no. 1, 2017

2017
[28]

Nonlinearsolve. jl: High-performance and ro- bust solvers for systems of nonlinear equations in julia,

A. Pal, F. Holtorf, A. Larsson, T. Loman, F. Schaefer, Q. Qu, A. Edel- man, C. Rackauckas, et al. , “Nonlinearsolve. jl: High-performance and ro- bust solvers for systems of nonlinear equations in julia,” arXiv preprint arXiv:2403.16341, 2024

2024 arXiv
[29]

Convex optimization in Julia,

M. Udell, K. Mohan, D. Zeng, J. Hong, S. Diamond, and S. Boyd, “Convex optimization in Julia,” SC14 Workshop on High Performance Technical Computing in Dynamic Languages , 2014. Appendix: Remaining Proofs Proof of Lemma 1. Introduce the difference between the inputs µ as µ = ˜...

2014

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Rate control for communica- tion networks: shadow prices, proportional fairness and stability,

F. P. Kelly, A. K. Maulloo, and D. K. H. Tan, “Rate control for communica- tion networks: shadow prices, proportional fairness and stability,” Journal of the Operational Research Society , vol. 49, pp. 237–252, Mar. 1998

work page 1998

[2] [2]

Optimization flow control. I. Basic algorithm and convergence,

S. Low and D. Lapsely, “Optimization flow control. I. Basic algorithm and convergence,” IEEE/ACM Transactions on Networking , vol. 7, pp. 861– 874, Dec. 1999

work page 1999

[3] [3]

Internet congestion control,

S. Low, F. Paganini, and J. Doyle, “Internet congestion control,” IEEE Control Systems Magazine , vol. 22, pp. 28–43, Feb. 2002

work page 2002

[4] [4]

Fairness and Stability of End-to-End Congestion Control*,

F. Kelly, “Fairness and Stability of End-to-End Congestion Control*,” Eu- ropean Journal of Control , vol. 9, pp. 159–176, Jan. 2003

work page 2003

[5] [5]

Optimal power flow pursuit,

E. Dall’Anese and A. Simonetto, “Optimal power flow pursuit,” IEEE Transactions on Smart Grid , vol. 9, no. 2, pp. 942–952, 2018

work page 2018

[6] [6]

A Survey of Distributed Optimization and Control Algorithms for Electric Power Systems,

D. K. Molzahn, F. D¨ orfler, H. Sandberg, S. H. Low, S. Chakrabarti, R. Baldick, and J. Lavaei, “A Survey of Distributed Optimization and Control Algorithms for Electric Power Systems,” IEEE Transactions on Smart Grid , vol. 8, pp. 2941–2962, Nov. 2017

work page 2017

[7] [7]

Deployment of an Online Feedback Optimization Controller for Reactive Power Flow Optimization in a Distribution Grid,

L. Ortmann, C. Rubin, A. Scozzafava, J. Lehmann, S. Bolognani, and F. D¨ orfler, “Deployment of an Online Feedback Optimization Controller for Reactive Power Flow Optimization in a Distribution Grid,” in 2023 IEEE PES Innovative Smart Grid Technologies Europe (ISGT EUROPE) , (Grenoble, France), pp. 1–6, IEEE, Oct. 2023

work page 2023

[8] [8]

A distributed algorithm for auto-balancing of hydronic cooling systems,

C. S. Kallesøe, B. K. Nielsen, A. Overgaard, and E. B. Sørensen, “A distributed algorithm for auto-balancing of hydronic cooling systems,” in 2019 IEEE Conference on Control Technology and Applications (CCTA) , pp. 655–660, 2019

work page 2019